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InDina [[probability theory]] andjeung [[statisticsstatistik]], the '''fungsi moment-generating function''' of atina [[randomvariabel variablerandom]] ''X'' isnyaeta
:<math>M_X(t)=E\left(e^{tX}\right).</math>
TheFungsi moment-generating function generates thengahasilkeun [[moment (mathematics)|moments]] of thetina [[probability distribution]], as followsnyaeta:
:<math>E\left(X^n\right)=M_X^{(n)}(0)=\left.\frac{\mathrm{d}^n}{\mathrm{d}t^n}\right|_{t=0} M_X(t).</math>
IfLamun ''X'' has a continousmibanda [[probability density function]] kontinyu ''f''(''x'') thenmangka thefungsi moment generating function is givendiberekeun byku
:<math>M_X(t) = \int_{-\infty}^\infty e^{tx} f(x)\,\mathrm{d}x</math>
:::<math> = 1 + tm_1 + \frac{t^2m_2}{2!} +\cdots,</math>
wheredimana <math>m_i</math> is the ''i''thngarupakeun [[moment (mathematics)|moment]] ka-''i''.
RegardlessTeu ofpaduli whetherkana [[probability distribution]] iskontinyu continuousatawa or notheunteu, thefungsi moment-generating function is given bydiberekeun theku [[Riemann-Stieltjes integral]]
:<math>\int_{-\infty}^\infty e^{tx}\,dF(x)</math>
wheredimana ''F'' is thenyaeta [[cumulative distribution function]].
RelatedKonsep conceptsnu includepakait thekaasup [[characteristic function]], the [[probability-generating function]], andjeung thefungsi [[cumulant]]-generating function. TheFungsi cumulant-generating functionnyaeta isbentuk thelogaritma logarithmtina of thefungsi moment-generating function.
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